
It is easy to see that \(\delta _\mathsf {rRB} (\mathrm {im~} (i)) \subset \mathrm {im~} (i)\) . Hence the differential \(\delta _\mathsf {rRB}\)  restricts to a differential (which we denote by \(\delta _\mathsf {RB}\) ) on \(C^\bullet _\mathsf {RB}(A,R;M, R_M)\) .
In other words, the complex \(\lbrace  C^\bullet _\mathsf {RB} (A, R; M, R_M), \delta _\mathsf {RB} \rbrace \)  is a subcomplex of \(\lbrace  C^\bullet _\mathsf {rRB} (A \xrightarrow{} A; M \xrightarrow{} M), \delta _\mathsf {rRB} \rbrace \) . The cohomology groups of \(\lbrace  C^\bullet _\mathsf {RB} (A, R; M, R_M), \delta _\mathsf {RB} \rbrace \)  are called the cohomology of the Rota-Baxter algebra \((A, R)\)  with coefficients in the Rota-Baxter bimodule \((M, R_M)\) . Note that the above cochain complex coincides with the one given in [1]}. Hence the cohomologies are also same.
